Mathematical Psychology and Psychophysiology

By Stephen Grossberg | Go to book overview
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TABLE 3
Viability, nurturance, and asymptotic frequency for k = -1
Genotypeuvw Frequency
A2A2 (altruistic) 1/9 3 1/3 1/16
A1A1 (neutral) 1 1 1 9/16
A1A2 (selfish) 16/9 3/4 4/3 6/16

Since k< 2, the selfish genotype should be favored, but, again, it is asymptotically less prevalent than the neutral genotype. For further consideration of "∣k> 1/r," in the context of other models, see [13] and [14].

5. Genes and genotypes. Fitnesses of genes can be defined by averaging over genotypes in which they occur. In the classical case, the fitnesses of A1 and A2 are
w1. = w11pn + w12qn and w2. = w12pn + w22qn.

Although (4.2) and (4.3) do not imply positive selection of genotypes with above average fitness, they do imply positive selection of genes with above average fitness. Thus discussions of evolution in terms of genes rather than genotypes avoid the pitfall which this paper has been concerned.

Many discussions of evolution (e.g., parts of [15]) are, in fact, cast in terms of genes. A limitation of this approach is that the fitness of a gene is not as intuitive as the fitness of a genotype. It is frequency dependent, and, if the heterozygote is most (or least) fit, even the ordering of w1. and w2. is different for large and small pn. Discussions that ignore this frequency dependence implicitly (and perhaps unconsciously) rule out heterozygote superiority, which is the basis for the example considered repeatedly in this paper.


REFERENCES

1. M. F. Norman, A "psychological" proof that certain Markov semigroups preserve differentiability, these PROCEEDINGS.

2. E. O. Wilson, Sociobiology: the new synthesis, Harvard Univ. Press, Cambridge, Mass., 1975.

3. W. J. Ewens, Mathematical population genetics, Springer-Verlag, New York, 1979.

4. S. J. Gould, Ever since Darwin: reflections in natural history, Norton, New York, 1977.

5. J. Maynard Smith, "Evolution and the theory of games", Amer. Sci. 64 ( 1976), 41-45.

6. G. F. Oster and S. M. Rocklin, "Optimization models in evolutionary biology", Some Mathematical Questions in Biology, vol. 10 ( S. A. Levin, ed.), Amer. Math. Soc., Providence, R. I., 1979.

7. P. D. Taylor and L. B. Jonker, "Evolutionarily stable strategies and game dynamics", Math. Biosci. 40 ( 1978), 145-156.

8. J. Hofbauer, P. Schuster and K. Sigmund, "A note on evolutionary stable strategies and game dynamics", J. Theoret. Biol. 81 ( 1979), 609-612.

9. W. F. Bodmer, "Differential fertility in population genetics models", Genetics 51 ( 1965), 411-424.

10. T. Nagylaki, Selection in one- and two-locus systems, Springer-Verlag, New York, 1977.

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